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Theorem addid1 9008
Description:  0 is an additive identity. This used to be one of our complex number axioms, until it was found to be dependent on the others. Based on ideas by Eric Schmidt. (Contributed by Scott Fenton, 3-Jan-2013.) (Proof shortened by Mario Carneiro, 27-May-2016.)
Assertion
Ref Expression
addid1  |-  ( A  e.  CC  ->  ( A  +  0 )  =  A )

Proof of Theorem addid1
Dummy variables  c  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 1re 8853 . 2  |-  1  e.  RR
2 ax-rnegex 8824 . 2  |-  ( 1  e.  RR  ->  E. c  e.  RR  ( 1  +  c )  =  0 )
3 ax-1ne0 8822 . . . . . 6  |-  1  =/=  0
4 oveq2 5882 . . . . . . . . . 10  |-  ( c  =  0  ->  (
1  +  c )  =  ( 1  +  0 ) )
54eqeq1d 2304 . . . . . . . . 9  |-  ( c  =  0  ->  (
( 1  +  c )  =  0  <->  (
1  +  0 )  =  0 ) )
65biimpcd 215 . . . . . . . 8  |-  ( ( 1  +  c )  =  0  ->  (
c  =  0  -> 
( 1  +  0 )  =  0 ) )
7 oveq2 5882 . . . . . . . . 9  |-  ( ( 1  +  0 )  =  0  ->  (
( ( _i  x.  _i )  x.  (
_i  x.  _i )
)  x.  ( 1  +  0 ) )  =  ( ( ( _i  x.  _i )  x.  ( _i  x.  _i ) )  x.  0 ) )
8 ax-icn 8812 . . . . . . . . . . . . . . 15  |-  _i  e.  CC
98, 8mulcli 8858 . . . . . . . . . . . . . 14  |-  ( _i  x.  _i )  e.  CC
109, 9mulcli 8858 . . . . . . . . . . . . 13  |-  ( ( _i  x.  _i )  x.  ( _i  x.  _i ) )  e.  CC
11 ax-1cn 8811 . . . . . . . . . . . . 13  |-  1  e.  CC
12 0cn 8847 . . . . . . . . . . . . 13  |-  0  e.  CC
1310, 11, 12adddii 8863 . . . . . . . . . . . 12  |-  ( ( ( _i  x.  _i )  x.  ( _i  x.  _i ) )  x.  ( 1  +  0 ) )  =  ( ( ( ( _i  x.  _i )  x.  ( _i  x.  _i ) )  x.  1 )  +  ( ( ( _i  x.  _i )  x.  ( _i  x.  _i ) )  x.  0 ) )
1410mulid1i 8855 . . . . . . . . . . . . 13  |-  ( ( ( _i  x.  _i )  x.  ( _i  x.  _i ) )  x.  1 )  =  ( ( _i  x.  _i )  x.  ( _i  x.  _i ) )
15 mul01 9007 . . . . . . . . . . . . . . 15  |-  ( ( ( _i  x.  _i )  x.  ( _i  x.  _i ) )  e.  CC  ->  ( (
( _i  x.  _i )  x.  ( _i  x.  _i ) )  x.  0 )  =  0 )
1610, 15ax-mp 8 . . . . . . . . . . . . . 14  |-  ( ( ( _i  x.  _i )  x.  ( _i  x.  _i ) )  x.  0 )  =  0
17 ax-i2m1 8821 . . . . . . . . . . . . . 14  |-  ( ( _i  x.  _i )  +  1 )  =  0
1816, 17eqtr4i 2319 . . . . . . . . . . . . 13  |-  ( ( ( _i  x.  _i )  x.  ( _i  x.  _i ) )  x.  0 )  =  ( ( _i  x.  _i )  +  1 )
1914, 18oveq12i 5886 . . . . . . . . . . . 12  |-  ( ( ( ( _i  x.  _i )  x.  (
_i  x.  _i )
)  x.  1 )  +  ( ( ( _i  x.  _i )  x.  ( _i  x.  _i ) )  x.  0 ) )  =  ( ( ( _i  x.  _i )  x.  (
_i  x.  _i )
)  +  ( ( _i  x.  _i )  +  1 ) )
2013, 19eqtri 2316 . . . . . . . . . . 11  |-  ( ( ( _i  x.  _i )  x.  ( _i  x.  _i ) )  x.  ( 1  +  0 ) )  =  ( ( ( _i  x.  _i )  x.  (
_i  x.  _i )
)  +  ( ( _i  x.  _i )  +  1 ) )
2120, 16eqeq12i 2309 . . . . . . . . . 10  |-  ( ( ( ( _i  x.  _i )  x.  (
_i  x.  _i )
)  x.  ( 1  +  0 ) )  =  ( ( ( _i  x.  _i )  x.  ( _i  x.  _i ) )  x.  0 )  <->  ( ( ( _i  x.  _i )  x.  ( _i  x.  _i ) )  +  ( ( _i  x.  _i )  +  1 ) )  =  0 )
2210, 9, 11addassi 8861 . . . . . . . . . . . 12  |-  ( ( ( ( _i  x.  _i )  x.  (
_i  x.  _i )
)  +  ( _i  x.  _i ) )  +  1 )  =  ( ( ( _i  x.  _i )  x.  ( _i  x.  _i ) )  +  ( ( _i  x.  _i )  +  1 ) )
239mulid1i 8855 . . . . . . . . . . . . . . 15  |-  ( ( _i  x.  _i )  x.  1 )  =  ( _i  x.  _i )
2423oveq2i 5885 . . . . . . . . . . . . . 14  |-  ( ( ( _i  x.  _i )  x.  ( _i  x.  _i ) )  +  ( ( _i  x.  _i )  x.  1
) )  =  ( ( ( _i  x.  _i )  x.  (
_i  x.  _i )
)  +  ( _i  x.  _i ) )
259, 9, 11adddii 8863 . . . . . . . . . . . . . . 15  |-  ( ( _i  x.  _i )  x.  ( ( _i  x.  _i )  +  1 ) )  =  ( ( ( _i  x.  _i )  x.  ( _i  x.  _i ) )  +  ( ( _i  x.  _i )  x.  1 ) )
2617oveq2i 5885 . . . . . . . . . . . . . . . 16  |-  ( ( _i  x.  _i )  x.  ( ( _i  x.  _i )  +  1 ) )  =  ( ( _i  x.  _i )  x.  0
)
27 mul01 9007 . . . . . . . . . . . . . . . . 17  |-  ( ( _i  x.  _i )  e.  CC  ->  (
( _i  x.  _i )  x.  0 )  =  0 )
289, 27ax-mp 8 . . . . . . . . . . . . . . . 16  |-  ( ( _i  x.  _i )  x.  0 )  =  0
2926, 28eqtri 2316 . . . . . . . . . . . . . . 15  |-  ( ( _i  x.  _i )  x.  ( ( _i  x.  _i )  +  1 ) )  =  0
3025, 29eqtr3i 2318 . . . . . . . . . . . . . 14  |-  ( ( ( _i  x.  _i )  x.  ( _i  x.  _i ) )  +  ( ( _i  x.  _i )  x.  1
) )  =  0
3124, 30eqtr3i 2318 . . . . . . . . . . . . 13  |-  ( ( ( _i  x.  _i )  x.  ( _i  x.  _i ) )  +  ( _i  x.  _i ) )  =  0
3231oveq1i 5884 . . . . . . . . . . . 12  |-  ( ( ( ( _i  x.  _i )  x.  (
_i  x.  _i )
)  +  ( _i  x.  _i ) )  +  1 )  =  ( 0  +  1 )
3322, 32eqtr3i 2318 . . . . . . . . . . 11  |-  ( ( ( _i  x.  _i )  x.  ( _i  x.  _i ) )  +  ( ( _i  x.  _i )  +  1
) )  =  ( 0  +  1 )
34 00id 9003 . . . . . . . . . . . 12  |-  ( 0  +  0 )  =  0
3534eqcomi 2300 . . . . . . . . . . 11  |-  0  =  ( 0  +  0 )
3633, 35eqeq12i 2309 . . . . . . . . . 10  |-  ( ( ( ( _i  x.  _i )  x.  (
_i  x.  _i )
)  +  ( ( _i  x.  _i )  +  1 ) )  =  0  <->  ( 0  +  1 )  =  ( 0  +  0 ) )
37 0re 8854 . . . . . . . . . . 11  |-  0  e.  RR
38 readdcan 9002 . . . . . . . . . . 11  |-  ( ( 1  e.  RR  /\  0  e.  RR  /\  0  e.  RR )  ->  (
( 0  +  1 )  =  ( 0  +  0 )  <->  1  = 
0 ) )
391, 37, 37, 38mp3an 1277 . . . . . . . . . 10  |-  ( ( 0  +  1 )  =  ( 0  +  0 )  <->  1  = 
0 )
4021, 36, 393bitri 262 . . . . . . . . 9  |-  ( ( ( ( _i  x.  _i )  x.  (
_i  x.  _i )
)  x.  ( 1  +  0 ) )  =  ( ( ( _i  x.  _i )  x.  ( _i  x.  _i ) )  x.  0 )  <->  1  =  0 )
417, 40sylib 188 . . . . . . . 8  |-  ( ( 1  +  0 )  =  0  ->  1  =  0 )
426, 41syl6 29 . . . . . . 7  |-  ( ( 1  +  c )  =  0  ->  (
c  =  0  -> 
1  =  0 ) )
4342necon3d 2497 . . . . . 6  |-  ( ( 1  +  c )  =  0  ->  (
1  =/=  0  -> 
c  =/=  0 ) )
443, 43mpi 16 . . . . 5  |-  ( ( 1  +  c )  =  0  ->  c  =/=  0 )
45 ax-rrecex 8825 . . . . 5  |-  ( ( c  e.  RR  /\  c  =/=  0 )  ->  E. x  e.  RR  ( c  x.  x
)  =  1 )
4644, 45sylan2 460 . . . 4  |-  ( ( c  e.  RR  /\  ( 1  +  c )  =  0 )  ->  E. x  e.  RR  ( c  x.  x
)  =  1 )
47 simpr 447 . . . . . . . . . 10  |-  ( ( ( ( c  e.  RR  /\  ( 1  +  c )  =  0 )  /\  (
x  e.  RR  /\  ( c  x.  x
)  =  1 ) )  /\  A  e.  CC )  ->  A  e.  CC )
48 simplrl 736 . . . . . . . . . . 11  |-  ( ( ( ( c  e.  RR  /\  ( 1  +  c )  =  0 )  /\  (
x  e.  RR  /\  ( c  x.  x
)  =  1 ) )  /\  A  e.  CC )  ->  x  e.  RR )
4948recnd 8877 . . . . . . . . . 10  |-  ( ( ( ( c  e.  RR  /\  ( 1  +  c )  =  0 )  /\  (
x  e.  RR  /\  ( c  x.  x
)  =  1 ) )  /\  A  e.  CC )  ->  x  e.  CC )
5047, 49mulcld 8871 . . . . . . . . 9  |-  ( ( ( ( c  e.  RR  /\  ( 1  +  c )  =  0 )  /\  (
x  e.  RR  /\  ( c  x.  x
)  =  1 ) )  /\  A  e.  CC )  ->  ( A  x.  x )  e.  CC )
51 simplll 734 . . . . . . . . . 10  |-  ( ( ( ( c  e.  RR  /\  ( 1  +  c )  =  0 )  /\  (
x  e.  RR  /\  ( c  x.  x
)  =  1 ) )  /\  A  e.  CC )  ->  c  e.  RR )
5251recnd 8877 . . . . . . . . 9  |-  ( ( ( ( c  e.  RR  /\  ( 1  +  c )  =  0 )  /\  (
x  e.  RR  /\  ( c  x.  x
)  =  1 ) )  /\  A  e.  CC )  ->  c  e.  CC )
5312a1i 10 . . . . . . . . 9  |-  ( ( ( ( c  e.  RR  /\  ( 1  +  c )  =  0 )  /\  (
x  e.  RR  /\  ( c  x.  x
)  =  1 ) )  /\  A  e.  CC )  ->  0  e.  CC )
5450, 52, 53adddid 8875 . . . . . . . 8  |-  ( ( ( ( c  e.  RR  /\  ( 1  +  c )  =  0 )  /\  (
x  e.  RR  /\  ( c  x.  x
)  =  1 ) )  /\  A  e.  CC )  ->  (
( A  x.  x
)  x.  ( c  +  0 ) )  =  ( ( ( A  x.  x )  x.  c )  +  ( ( A  x.  x )  x.  0 ) ) )
5511a1i 10 . . . . . . . . . . . . 13  |-  ( ( ( ( c  e.  RR  /\  ( 1  +  c )  =  0 )  /\  (
x  e.  RR  /\  ( c  x.  x
)  =  1 ) )  /\  A  e.  CC )  ->  1  e.  CC )
5655, 52, 53addassd 8873 . . . . . . . . . . . 12  |-  ( ( ( ( c  e.  RR  /\  ( 1  +  c )  =  0 )  /\  (
x  e.  RR  /\  ( c  x.  x
)  =  1 ) )  /\  A  e.  CC )  ->  (
( 1  +  c )  +  0 )  =  ( 1  +  ( c  +  0 ) ) )
57 simpllr 735 . . . . . . . . . . . . 13  |-  ( ( ( ( c  e.  RR  /\  ( 1  +  c )  =  0 )  /\  (
x  e.  RR  /\  ( c  x.  x
)  =  1 ) )  /\  A  e.  CC )  ->  (
1  +  c )  =  0 )
5857oveq1d 5889 . . . . . . . . . . . 12  |-  ( ( ( ( c  e.  RR  /\  ( 1  +  c )  =  0 )  /\  (
x  e.  RR  /\  ( c  x.  x
)  =  1 ) )  /\  A  e.  CC )  ->  (
( 1  +  c )  +  0 )  =  ( 0  +  0 ) )
5956, 58eqtr3d 2330 . . . . . . . . . . 11  |-  ( ( ( ( c  e.  RR  /\  ( 1  +  c )  =  0 )  /\  (
x  e.  RR  /\  ( c  x.  x
)  =  1 ) )  /\  A  e.  CC )  ->  (
1  +  ( c  +  0 ) )  =  ( 0  +  0 ) )
6034, 59, 573eqtr4a 2354 . . . . . . . . . 10  |-  ( ( ( ( c  e.  RR  /\  ( 1  +  c )  =  0 )  /\  (
x  e.  RR  /\  ( c  x.  x
)  =  1 ) )  /\  A  e.  CC )  ->  (
1  +  ( c  +  0 ) )  =  ( 1  +  c ) )
6137a1i 10 . . . . . . . . . . . 12  |-  ( ( ( ( c  e.  RR  /\  ( 1  +  c )  =  0 )  /\  (
x  e.  RR  /\  ( c  x.  x
)  =  1 ) )  /\  A  e.  CC )  ->  0  e.  RR )
6251, 61readdcld 8878 . . . . . . . . . . 11  |-  ( ( ( ( c  e.  RR  /\  ( 1  +  c )  =  0 )  /\  (
x  e.  RR  /\  ( c  x.  x
)  =  1 ) )  /\  A  e.  CC )  ->  (
c  +  0 )  e.  RR )
631a1i 10 . . . . . . . . . . 11  |-  ( ( ( ( c  e.  RR  /\  ( 1  +  c )  =  0 )  /\  (
x  e.  RR  /\  ( c  x.  x
)  =  1 ) )  /\  A  e.  CC )  ->  1  e.  RR )
64 readdcan 9002 . . . . . . . . . . 11  |-  ( ( ( c  +  0 )  e.  RR  /\  c  e.  RR  /\  1  e.  RR )  ->  (
( 1  +  ( c  +  0 ) )  =  ( 1  +  c )  <->  ( c  +  0 )  =  c ) )
6562, 51, 63, 64syl3anc 1182 . . . . . . . . . 10  |-  ( ( ( ( c  e.  RR  /\  ( 1  +  c )  =  0 )  /\  (
x  e.  RR  /\  ( c  x.  x
)  =  1 ) )  /\  A  e.  CC )  ->  (
( 1  +  ( c  +  0 ) )  =  ( 1  +  c )  <->  ( c  +  0 )  =  c ) )
6660, 65mpbid 201 . . . . . . . . 9  |-  ( ( ( ( c  e.  RR  /\  ( 1  +  c )  =  0 )  /\  (
x  e.  RR  /\  ( c  x.  x
)  =  1 ) )  /\  A  e.  CC )  ->  (
c  +  0 )  =  c )
6766oveq2d 5890 . . . . . . . 8  |-  ( ( ( ( c  e.  RR  /\  ( 1  +  c )  =  0 )  /\  (
x  e.  RR  /\  ( c  x.  x
)  =  1 ) )  /\  A  e.  CC )  ->  (
( A  x.  x
)  x.  ( c  +  0 ) )  =  ( ( A  x.  x )  x.  c ) )
6854, 67eqtr3d 2330 . . . . . . 7  |-  ( ( ( ( c  e.  RR  /\  ( 1  +  c )  =  0 )  /\  (
x  e.  RR  /\  ( c  x.  x
)  =  1 ) )  /\  A  e.  CC )  ->  (
( ( A  x.  x )  x.  c
)  +  ( ( A  x.  x )  x.  0 ) )  =  ( ( A  x.  x )  x.  c ) )
69 mul31 8996 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  x  e.  CC  /\  c  e.  CC )  ->  (
( A  x.  x
)  x.  c )  =  ( ( c  x.  x )  x.  A ) )
7047, 49, 52, 69syl3anc 1182 . . . . . . . . 9  |-  ( ( ( ( c  e.  RR  /\  ( 1  +  c )  =  0 )  /\  (
x  e.  RR  /\  ( c  x.  x
)  =  1 ) )  /\  A  e.  CC )  ->  (
( A  x.  x
)  x.  c )  =  ( ( c  x.  x )  x.  A ) )
71 simplrr 737 . . . . . . . . . 10  |-  ( ( ( ( c  e.  RR  /\  ( 1  +  c )  =  0 )  /\  (
x  e.  RR  /\  ( c  x.  x
)  =  1 ) )  /\  A  e.  CC )  ->  (
c  x.  x )  =  1 )
7271oveq1d 5889 . . . . . . . . 9  |-  ( ( ( ( c  e.  RR  /\  ( 1  +  c )  =  0 )  /\  (
x  e.  RR  /\  ( c  x.  x
)  =  1 ) )  /\  A  e.  CC )  ->  (
( c  x.  x
)  x.  A )  =  ( 1  x.  A ) )
7347mulid2d 8869 . . . . . . . . 9  |-  ( ( ( ( c  e.  RR  /\  ( 1  +  c )  =  0 )  /\  (
x  e.  RR  /\  ( c  x.  x
)  =  1 ) )  /\  A  e.  CC )  ->  (
1  x.  A )  =  A )
7470, 72, 733eqtrd 2332 . . . . . . . 8  |-  ( ( ( ( c  e.  RR  /\  ( 1  +  c )  =  0 )  /\  (
x  e.  RR  /\  ( c  x.  x
)  =  1 ) )  /\  A  e.  CC )  ->  (
( A  x.  x
)  x.  c )  =  A )
75 mul01 9007 . . . . . . . . 9  |-  ( ( A  x.  x )  e.  CC  ->  (
( A  x.  x
)  x.  0 )  =  0 )
7650, 75syl 15 . . . . . . . 8  |-  ( ( ( ( c  e.  RR  /\  ( 1  +  c )  =  0 )  /\  (
x  e.  RR  /\  ( c  x.  x
)  =  1 ) )  /\  A  e.  CC )  ->  (
( A  x.  x
)  x.  0 )  =  0 )
7774, 76oveq12d 5892 . . . . . . 7  |-  ( ( ( ( c  e.  RR  /\  ( 1  +  c )  =  0 )  /\  (
x  e.  RR  /\  ( c  x.  x
)  =  1 ) )  /\  A  e.  CC )  ->  (
( ( A  x.  x )  x.  c
)  +  ( ( A  x.  x )  x.  0 ) )  =  ( A  + 
0 ) )
7868, 77, 743eqtr3d 2336 . . . . . 6  |-  ( ( ( ( c  e.  RR  /\  ( 1  +  c )  =  0 )  /\  (
x  e.  RR  /\  ( c  x.  x
)  =  1 ) )  /\  A  e.  CC )  ->  ( A  +  0 )  =  A )
7978exp42 594 . . . . 5  |-  ( ( c  e.  RR  /\  ( 1  +  c )  =  0 )  ->  ( x  e.  RR  ->  ( (
c  x.  x )  =  1  ->  ( A  e.  CC  ->  ( A  +  0 )  =  A ) ) ) )
8079rexlimdv 2679 . . . 4  |-  ( ( c  e.  RR  /\  ( 1  +  c )  =  0 )  ->  ( E. x  e.  RR  ( c  x.  x )  =  1  ->  ( A  e.  CC  ->  ( A  +  0 )  =  A ) ) )
8146, 80mpd 14 . . 3  |-  ( ( c  e.  RR  /\  ( 1  +  c )  =  0 )  ->  ( A  e.  CC  ->  ( A  +  0 )  =  A ) )
8281rexlimiva 2675 . 2  |-  ( E. c  e.  RR  (
1  +  c )  =  0  ->  ( A  e.  CC  ->  ( A  +  0 )  =  A ) )
831, 2, 82mp2b 9 1  |-  ( A  e.  CC  ->  ( A  +  0 )  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459   E.wrex 2557  (class class class)co 5874   CCcc 8751   RRcr 8752   0cc0 8753   1c1 8754   _ici 8755    + caddc 8756    x. cmul 8758
This theorem is referenced by:  cnegex  9009  addid2  9011  addcan2  9013  addid1i  9015  addid1d  9028  subid  9083  subid1  9084  shftval3  11587  reim0  11619  isercolllem3  12156  fsumcvg  12201  summolem2a  12204  ovolicc1  18891  relexpadd  24050  brbtwn2  24604  axsegconlem1  24616  ax5seglem4  24631  axeuclid  24662  axcontlem2  24664  axcontlem4  24666  stoweidlem11  27862  stoweidlem26  27877
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-po 4330  df-so 4331  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-ltxr 8888
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